Calculate the percentage decrease in weight of a body when taken to a tunnel 35 km below the surface of earth.

Answer

To calculate the percentage decrease in weight of a body when taken to a tunnel 35 km below the surface of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the percentage decrease in weight when a body is taken 35 km below the Earth's surface.

2. **Formula for Acceleration Due to Gravity Below the Surface**: The acceleration due to gravity at a depth \( D \) below the surface of the Earth is given by the formula:
\[ g' = g \left(1 - \frac{D}{R}\right) \]
where:
- \( g' \) = acceleration due to gravity at depth \( D \)
- \( g \) = acceleration due to gravity at the surface of the Earth
- \( D \) = depth below the surface (35 km in this case)
- \( R \) = radius of the Earth (approximately 6400 km)

3. **Substituting Values**: We know \( D = 35 \) km and \( R = 6400 \) km. Therefore, we can substitute these values into the formula:
\[ g' = g \left(1 - \frac{35}{6400}\right) \]

4. **Calculate the Ratio**: Calculate the fraction:
\[ \frac{35}{6400} = 0.00546875 \]

5. **Calculate \( g' \)**: Now substitute this back into the formula:
\[ g' = g \left(1 - 0.00546875\right) = g \times 0.99453125 \]

6. **Weight Calculation**: The weight of a body is given by \( W = mg \). Thus, the initial weight at the surface is:
\[ W_{\text{initial}} = mg \]
The weight at depth \( D \) is:
\[ W_{\text{final}} = mg' \]

7. **Percentage Decrease in Weight**: The percentage decrease in weight can be calculated using the formula:
\[ \text{Percentage Decrease} = \frac{W_{\text{initial}} - W_{\text{final}}}{W_{\text{initial}}} \times 100 \]
Substituting the weights:
\[ \text{Percentage Decrease} = \frac{mg - mg'}{mg} \times 100 = \frac{g - g'}{g} \times 100 \]

8. **Substituting \( g' \)**: Now substitute \( g' \):
\[ \text{Percentage Decrease} = \frac{g - g \times 0.99453125}{g} \times 100 = (1 - 0.99453125) \times 100 \]

9. **Calculate the Final Value**:
\[ \text{Percentage Decrease} = 0.00546875 \times 100 = 0.546875\% \]

10. **Final Answer**: Rounding to two decimal places, the percentage decrease in weight is approximately:
\[ \text{Percentage Decrease} \approx 0.55\% \]